Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Approximate D-optimal design and equilibrium measure

Didier Henrion, Jean Bernard Lasserre

TL;DR

This work introduces a boundary-aware variant of the D-optimal design problem, incorporating an information matrix that reflects the geometry of the design space via generators $G$ and boundary polynomials. For the unit ball, unit box, and canonical simplex in any dimension, the equilibrium measure $\phi^*$ from pluripotential theory is shown to be the optimal solution for all degrees $n$, turning design construction into finding cubatures of $\phi^*$ exact up to degree $2n$, with the atomic designs $\nu^*_n$ converging to $\phi^*$ in the weak-* sense. The paper connects these optimal designs to Fekete/Fejér configurations and extends the framework to general compact basic semi-algebraic sets, leveraging a two-step moment-SOS relaxation that remains exact under suitable assumptions. Practically, this bridges approximation theory and optimal experimental design, offering explicit pathways to compute designs via known cubatures (e.g., Gauss–Chebyshev on the box) or via adapted SOS-based steps for more complex spaces, and it highlights potential for broad applicability in geometry-aware experimental design.

Abstract

We introduce a minor variant of the approximate D-optimal design of experiments with a more general information matrix that takes into account the representation of the design space S. The main motivation (and result) is that if S in R^d is the unit ball, the unit box or the canonical simplex, then remarkably, for every dimension d and every degree n, one obtains an optimal solution in closed form, namely the equilibrium measure of S (in pluripotential theory). Equivalently, for each degree n, the unique optimal solution is the vector of moments (up to degree 2n) of the equilibrium measure of S. Hence finding an optimal design reduces to finding a cubature for the equilibrium measure, with atoms in S, positive weights, and exact up to degree 2n. In addition, any resulting sequence of atomic D-optimal measures converges to the equilibrium measure of S for the weak-star topology, as n increases. Links with Fekete sets of points are also discussed. More general compact basic semi-algebraic sets are also considered, and a previously developed two-step design algorithm is easily adapted to this new variant of D-optimal design problem.

Approximate D-optimal design and equilibrium measure

TL;DR

This work introduces a boundary-aware variant of the D-optimal design problem, incorporating an information matrix that reflects the geometry of the design space via generators and boundary polynomials. For the unit ball, unit box, and canonical simplex in any dimension, the equilibrium measure from pluripotential theory is shown to be the optimal solution for all degrees , turning design construction into finding cubatures of exact up to degree , with the atomic designs converging to in the weak-* sense. The paper connects these optimal designs to Fekete/Fejér configurations and extends the framework to general compact basic semi-algebraic sets, leveraging a two-step moment-SOS relaxation that remains exact under suitable assumptions. Practically, this bridges approximation theory and optimal experimental design, offering explicit pathways to compute designs via known cubatures (e.g., Gauss–Chebyshev on the box) or via adapted SOS-based steps for more complex spaces, and it highlights potential for broad applicability in geometry-aware experimental design.

Abstract

We introduce a minor variant of the approximate D-optimal design of experiments with a more general information matrix that takes into account the representation of the design space S. The main motivation (and result) is that if S in R^d is the unit ball, the unit box or the canonical simplex, then remarkably, for every dimension d and every degree n, one obtains an optimal solution in closed form, namely the equilibrium measure of S (in pluripotential theory). Equivalently, for each degree n, the unique optimal solution is the vector of moments (up to degree 2n) of the equilibrium measure of S. Hence finding an optimal design reduces to finding a cubature for the equilibrium measure, with atoms in S, positive weights, and exact up to degree 2n. In addition, any resulting sequence of atomic D-optimal measures converges to the equilibrium measure of S for the weak-star topology, as n increases. Links with Fekete sets of points are also discussed. More general compact basic semi-algebraic sets are also considered, and a previously developed two-step design algorithm is easily adapted to this new variant of D-optimal design problem.
Paper Structure (15 sections, 3 theorems, 68 equations, 3 figures)

This paper contains 15 sections, 3 theorems, 68 equations, 3 figures.

Table of Contents

  1. Introduction
  2. Contribution
  3. Notation, definitions and preliminaries
  4. Notation and definitions
  5. Moment and localizing matrix
  6. Christoffel-Darboux kernel and Christoffel function
  7. Equilibrium measure
  8. Background on approximate D-optimal design
  9. Convex relaxations
  10. Main result
  11. Comparison with the original formulation
  12. Computing an optimal atomic measure
  13. More general semi-algebraic sets
  14. Basic semi-algebraic sets $S$
  15. Conclusion

Key Result

Theorem 1

Let $n\in\mathbb{N}$ be fixed, arbitrary, and $S^{\star}=S^{O}$, $S^{\square}$, or $S^{\triangle}$ (fixed). Then (i) The equilibrium measure $\phi^{\star}$ of $S^{\star}$ is an optimal solution of def-new, the optimal moment matrices $\mathbf{M}_{n-d_g}(g\,\phi^{\star})$, $g\in G^\star_n$, are uniqu (iii) There are $s_n\leq m\leq s_{2n}$ distinct points $\mathbf{x}^*_j\in S^\star$, and a positive

Figures (3)

  • Figure 1: Degree 4 D-optimal design on the disk solving problem \ref{['def-D-optimal']}. The optimal points (red) are located on the CD polynomial unit level set (blue).
  • Figure 2: Degree 4 D-optimal design on the disk (black) with points (red) solving variant problem \ref{['D-variant']}.
  • Figure 3: Degree 16 D-optimal design on unit interval: CD polynomial unit level set points (red) solve problem \ref{['def-D-optimal']}, whereas roots of the degree 9 Chebyshev polnomials (blue) solve variant problem \ref{['D-variant']}.

Theorems & Definitions (7)

  • Theorem 1
  • Example 1
  • Example 2
  • Corollary 1
  • Example 3
  • Theorem 2
  • Remark 1